Minors and Cofactors

Expansion by minors, introduced informally in the previous section, is stated precisely using two related ideas: the minor and the cofactor of an entry.

The minor of the entry in row $i$, column $j$ of a determinant, written $M_{ij}$, is the determinant that remains after deleting row $i$ and column $j$ entirely. For $A=\begin{pmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{pmatrix},$ the minor of $b_2$ (row 2, column 2) is $M_{22} = \begin{vmatrix} a_1 & c_1 \\ a_3 & c_3 \end{vmatrix},$ obtained by striking out row 2 and column 2.

The cofactor of the same entry, written $C_{ij}$, attaches the correct sign to the minor: $C_{ij} = (-1)^{i+j} M_{ij}.$ Whenever $i+j$ is even the sign is $+$; whenever $i+j$ is odd the sign is $-$ — this is exactly the alternating pattern given in the previous section, now stated as a formula rather than a picture. So $C_{22} = (-1)^{2+2}M_{22} = +M_{22}$, while $C_{21} = (-1)^{2+1}M_{21} = -M_{21}$. …